The Petersson adjoint on good Hecke eigenforms

On S_k(N, χ) the Petersson adjoint of the Hecke operator Tₚ at a prime p ∤ N is χ(p)⁻¹ Tₚ. Evaluated on good Hecke eigenforms this gives two consequences for their eigenvalues.

Reality up to the nebentypus. A good Hecke eigenform pairs nontrivially with itself, so its eigenvalue λ_p is fixed by c ↦ conj (χ(p)⁻¹ c), that is λ_p = χ(p) · conj λ_p.

Orthogonality. Good Hecke eigenforms whose eigenvalues differ at a prime p ∤ N are orthogonal: for a common nebentypus χ because of the adjoint relation, and otherwise because the nebentypus decomposition is orthogonal.

Main results

• HeckeRing.GL2.EigenformAwayFromLevel.eigenvalue_eq_mul_conj: at a good prime p, the eigenvalue of a good Hecke eigenform satisfies λ_p = χ(p) · conj λ_p.
• HeckeRing.GL2.EigenformAwayFromLevel.peterssonInnerCosets_eq_zero_of_eigenvalue_ne: good Hecke eigenforms with distinct eigenvalues at a good prime are orthogonal.

References

• T. Miyake, Modular forms, Theorem 4.6.13(2).
• F. Diamond and J. Shurman, A first course in modular forms, Theorem 5.8.2.

theorem HeckeRing.GL2.EigenformAwayFromLevel.heckeRingHomCuspCharSpace_heckeTGeneratorGamma0_eq_smul {N : ℕ} [NeZero N] {k : ℤ} {χ : (ZMod N)ˣ →* ℂˣ} {f : EigenformAwayFromLevel N k} {x : ↥(cuspFormCharSpace k χ)} (hχ : f.χ = χ) (hx : f.toCuspForm = ↑x) {p : ℕ} (hp : Nat.Prime p) (hpN : p.Coprime N) : ((heckeRingHomCuspCharSpace k χ) (heckeTGeneratorGamma0 N p)) x = f.eigenvalue ⟨p, ⋯⟩ hpN • x

A good Hecke eigenform of nebentypus χ, seen in S_k(N, χ), is an eigenvector of the Hecke ring generator Tₚ at every prime p not dividing N, with its eigenvalue at p.

theorem HeckeRing.GL2.EigenformAwayFromLevel.eigenvalue_eq_mul_conj {N : ℕ} [NeZero N] {k : ℤ} (f : EigenformAwayFromLevel N k) {p : ℕ} (hp : Nat.Prime p) (hpN : p.Coprime N) : f.eigenvalue ⟨p, ⋯⟩ hpN = ↑(f.χ (ZMod.unitOfCoprime p hpN)) * (starRingEnd ℂ) (f.eigenvalue ⟨p, ⋯⟩ hpN)

The eigenvalues of a good Hecke eigenform are real up to the nebentypus: at a prime p ∤ N, the eigenvalue λ_p of a good Hecke eigenform of nebentypus χ satisfies λ_p = χ(p) · conj λ_p. This is the shadow of the Petersson adjoint Tₚ* = χ(p)⁻¹ Tₚ on the eigenvector; for χ(p) = 1, for instance for trivial nebentypus, it says that λ_p is real.

theorem HeckeRing.GL2.EigenformAwayFromLevel.peterssonInnerCosets_eq_zero_of_eigenvalue_ne {N : ℕ} [NeZero N] {k : ℤ} {f g : EigenformAwayFromLevel N k} {p : ℕ} (hp : Nat.Prime p) (hpN : p.Coprime N) (hne : f.eigenvalue ⟨p, ⋯⟩ hpN ≠ g.eigenvalue ⟨p, ⋯⟩ hpN) : f.peterssonInnerCosets g.toCuspForm = 0

Good Hecke eigenforms with distinct eigenvalues are orthogonal. Two good Hecke eigenforms whose eigenvalues differ at a prime p not dividing N are Petersson-orthogonal.